# Quantum Bayesian Networks

## January 4, 2010

### I Don’t Have to Show You Any Stinking Mixed States

Filed under: Uncategorized — rrtucci @ 2:41 pm

From the 1948 film “The Treasure of the Sierra Madre” with Humphrey Bogart

I’ve written several papers (and a computer program called Quantum Fog) on quantum bayesian networks. QB nets are an adaptation of the usual Bayesian networks to quantum mechanics. Quantum computerists can think of QB nets as an alternative way of drawing quantum circuits.

QB nets portray the evolution of pure states. A little knowledge is a dangerous thing, or, at least, an annoying thing. Many beginners immediately reject QB nets because QB nets cannot handle mixed states. To which I reply, as General McAuliffe once did at Bastogne under a similarly annoying threat: “Nuts!”

QB nets don’t need any stinking mixed states. Given a mixed state, you can always replace it by a pure state (called a “purification”) that lives in a bigger space. Technically, if a mixed state is described by a density matrix $\rho_1$ acting on a Hilbert space ${\cal H}_1$, you can always find a Hilbert space ${\cal H}_2$ and a pure state $|\Psi_{12}\rangle\in {\cal H}_1\otimes{\cal H}_2$ such that $\rho_1= tr_2(|\Psi_{12}\rangle\langle\Psi_{12}|)$. For example, if $\rho_1$ has eigenvalues $w_r$ and eigenvectors $|\psi_r\rangle$ so that $\rho_1 = \sum_r w_r |\psi_r\rangle\langle\psi_r|$, then a possible purification of $\rho_1$ is $|\Psi_{12}\rangle = \sum_r \sqrt{w_r} |\psi_r\rangle\otimes |\psi_r\rangle$.